Parity-sampler status of the (x, y, xy) hypergraph

Determine whether the non-regular 3-uniform hypergraph formed by triples (x, y, xy) over a finite group G can be used as a parity sampler, despite its lack of regularity.

Background

In analyzing averages involving products of the form f1(x) f2(y) f3(xy) with x sampled from the group G and y sampled from a subset S, the authors consider the combinatorial structure underlying these queries: hyperedges given by triples (x, y, xy).

They note that the resulting hypergraph is not regular, raising uncertainty about whether such a structure can support parity-sampling properties typically leveraged in derandomized testing and PCP constructions. The explicit uncertainty is stated in the Hypergraph paragraph of the Product (x,y,xy) subsection.